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Pyth, 5 bytes
x1iLQ
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How it works
Note that Pyth uses 0-indexing.
x1iLQ Q = eval(input())
x1iLQQ implicit Q at the end
iLQQ [gcd(Q,0), gcd(Q,1), ..., gcd(Q,Q-1)]
x1 all occurences of 1 in the above list (return their indices)